ACT Math: Statistics
Statistics questions on the ACT test whether you understand what each measure actually describes, center or spread, rather than requiring heavy computation. This free, complete lesson covers mean, median, mode, and range, weighted averages, standard deviation and outliers, and quartiles, interquartile range, and box plots. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.
1. Mean, Median, Mode, and Range
The mean is the sum of all values divided by how many there are. The median is the middle value when the data is ordered (or the average of the two middle values, if there's an even count). The mode is the most frequently occurring value. The range is the largest value minus the smallest.
Worked Example: Find the mean, median, mode, and range of {4, 7, 7, 9, 13}.
Mean: (4 + 7 + 7 + 9 + 13) ÷ 5 = 40 ÷ 5 = 8.
Median (already ordered, 5 values): the middle value is 7.
Mode: 7 (it appears twice). Range: 13 − 4 = 9.
1. Find the mean of {12, 15, 18, 21, 24}.
Explanation: (12 + 15 + 18 + 21 + 24) ÷ 5 = 90 ÷ 5 = 18.
2. Find the median of {8, 3, 15, 10, 6}.
Explanation: Ordered: 3, 6, 8, 10, 15. The middle value is 8.
3. Find the median of {2, 9, 4, 7}.
Explanation: Ordered: 2, 4, 7, 9. With an even count, average the two middle values: (4 + 7) ÷ 2 = 5.5.
4. Find the mode of {6, 3, 6, 8, 3, 6, 9}.
Explanation: 6 appears three times, more than any other value.
5. Find the range of {45, 12, 78, 23, 56}.
Explanation: 78 − 12 = 66.
2. Weighted Averages
A weighted average accounts for groups of different sizes contributing to an overall average, larger groups pull the average toward their own value more strongly than smaller groups do.
Worked Example: The monthly fees for single rooms at 5 colleges are $370, $310, $340, $380, and $310. Find the mean.
(370 + 310 + 340 + 380 + 310) ÷ 5 = 1,710 ÷ 5 = 342
1. In a class, 10 students scored an average of 90 on a quiz, and 15 students scored an average of 80. What is the average score for the entire class of 25 students?
Explanation: (10 × 90 + 15 × 80) ÷ 25 = (900 + 1,200) ÷ 25 = 2,100 ÷ 25 = 84.
2. A student has three equally-weighted test scores of 85, 90, and 95, plus a final exam (worth twice the weight of one test) scored 80. What is the weighted average?
Explanation: Total weight = 1 + 1 + 1 + 2 = 5. Weighted sum = 85 + 90 + 95 + 80(2) = 430. Average: 430 ÷ 5 = 86.
3. A store sold 30 shirts at $20 each and 20 shirts at $15 each. What was the average price per shirt sold?
Explanation: (30 × 20 + 20 × 15) ÷ 50 = (600 + 300) ÷ 50 = 900 ÷ 50 = $18.
4. A class of 12 students has an average score of 75. If one more student joins with a score of 88, what is the new average for all 13 students?
Explanation: Original total: 12 × 75 = 900. New total: 900 + 88 = 988. New average: 988 ÷ 13 = 76.
3. Standard Deviation and Outliers
Standard deviation measures how spread out a data set is from its mean, more spread means a larger standard deviation; tightly clustered data means a smaller one. An outlier is a value far from the rest of the data; the mean is much more sensitive to outliers than the median.
Worked Example: Data set A has values clustered tightly around its mean; Data set B has values spread far apart. Which has the higher standard deviation?
Data set B, since its values deviate from the mean by more, on average, than data set A's values do.
1. Which of the following data sets would have the LOWEST standard deviation?
Explanation: Every value in {10, 10, 10, 10, 10} is identical, meaning there's no spread at all, and a standard deviation of exactly 0.
2. A data set of four salaries has a mean of $45,000 and a median of $40,000. A person earning $1,000,000 joins the group. Which measure is affected the most: the mean or the median?
Explanation: The mean is pulled dramatically toward extreme values like this outlier, while the median, based on position rather than magnitude, shifts only slightly.
3. For a normal distribution, approximately what percent of the data falls within 1 standard deviation of the mean?
Explanation: By the empirical rule, about 68% of normally distributed data falls within 1 standard deviation of the mean.
4. For a normal distribution, approximately what percent of the data falls within 2 standard deviations of the mean?
Explanation: By the empirical rule, about 95% of normally distributed data falls within 2 standard deviations of the mean.
5. If a data set's variance is 25, what is its standard deviation?
Explanation: Standard deviation is the square root of variance: √25 = 5.
4. Quartiles, Interquartile Range, and Box Plots
A box plot displays five values: the minimum, the first quartile (Q1, the 25th percentile), the median (Q2), the third quartile (Q3, the 75th percentile), and the maximum. The interquartile range (IQR) is Q3 − Q1, the spread of the middle 50% of the data.
Anatomy of a box plot
Worked Example: A box plot shows: min = 10, Q1 = 20, median = 30, Q3 = 45, max = 60. Find the interquartile range (IQR).
IQR = Q3 − Q1 = 45 − 20 = 25
1. A box plot shows: min = 5, Q1 = 15, median = 25, Q3 = 35, max = 50. Find the IQR.
Explanation: IQR = Q3 − Q1 = 35 − 15 = 20.
2. Using the same box plot, find the range.
Explanation: Range = max − min = 50 − 5 = 45.
3. What percentage of the data lies within the "box" of a box plot, between Q1 and Q3?
Explanation: Since Q1 marks the 25th percentile and Q3 marks the 75th, the box always contains the middle 50% of the data.
4. A box plot's median line sits noticeably closer to Q1 than to Q3. What does this suggest about the data's distribution?
Explanation: A median closer to Q1 means the upper 50% of the data (between the median and Q3) is more spread out than the lower 50%, a sign of right-skewed data.
5. A data set has Q1 = 18 and Q3 = 42. A value of 78 is being checked against the standard outlier rule (more than 1.5 × IQR above Q3). Is 78 an outlier by this rule?
Explanation: IQR = 42 − 18 = 24. The outlier threshold is Q3 + 1.5(IQR) = 42 + 36 = 78. Since 78 meets this threshold, it qualifies as an outlier by the standard rule.
Common Mistakes to Avoid
- Forgetting to order the data before finding the median. The median depends entirely on position, not order of appearance, always sort the values first.
- Averaging the two middle values incorrectly, or forgetting to average them at all. With an even number of data points, the median is the mean of the two middle values, not simply "the closest one."
- Treating a weighted average like a simple average. When group sizes differ, the overall average isn't just the average of the group averages, each group's size (its weight) matters.
- Assuming the median shifts as much as the mean when an outlier is added. The mean is highly sensitive to extreme values; the median usually barely moves.
- Confusing range with interquartile range. Range uses the absolute minimum and maximum (max − min); IQR uses only the middle 50% of the data (Q3 − Q1), and is far less sensitive to outliers.
Frequently Asked Questions
Why is the median often described as "resistant to outliers"?
The median only cares about which value sits in the middle position, not how far away the other values are. An extremely large or small value can shift which numbers are "in the middle," but usually only by a little, unlike the mean, which is dragged directly toward any extreme value.
Do I need to memorize the standard deviation formula for the ACT?
No. ACT questions test conceptual understanding, recognizing which data set is more spread out, or how standard deviation relates to the shape of a distribution, rather than requiring the full computation.
How is a box plot useful beyond just showing five numbers?
A box plot visually reveals a data set's spread and symmetry at a glance: a wide box means a larger IQR, and an off-center median line signals a skewed distribution, information that's harder to see from a plain list of numbers.
Where can I practice more problems like these?
The Statistics quizzes in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.